MathDB

2

a!b!c!d!=24!

Compute the number of ordered quadruples of positive integers

(a,b,c,d)

such that

a!b!c!d!=24!

<span class='latex-bold'>(A)</span>~4

<span class='latex-bold'>(B)</span>~4!

<span class='latex-bold'>(C)</span>~4^4

<span class='latex-bold'>(D)</span>~\text{None of the above}

taxicab number

What is the smallest positive integer

n

such that

n=x^3+y^3

for two different positive integer tuples

(x,y)

?

Problem 11

2

condition for z/(1-z^2)

z

is a complex number and

|z|=1

and

z^2\ne1

. Then

\frac z{1-z^2}

lies on

<span class='latex-bold'>(A)</span>~\text{a line not through origin}

<span class='latex-bold'>(B)</span>~\text{|z|=2}

<span class='latex-bold'>(C)</span>~x-\text{axis}

<span class='latex-bold'>(D)</span>~y-\text{axis}

given trigonometric equality, find angle

\angle A,\angle B,\angle C

are angles of a triangle such that

\sin^2A+\sin^2B=\sin^2C

, then

\angle C

in degrees is equal to

<span class='latex-bold'>(A)</span>~30

<span class='latex-bold'>(B)</span>~90

<span class='latex-bold'>(C)</span>~45

<span class='latex-bold'>(D)</span>~\text{none of the above}

Problem 10

2

# of solns. to 3^x+4^x=8^x over R

Number of solutions of the equation

3^x+4^x=8^x

in reals is

<span class='latex-bold'>(A)</span>~0

<span class='latex-bold'>(B)</span>~1

<span class='latex-bold'>(C)</span>~2

<span class='latex-bold'>(D)</span>~\infty

number of nondecreasing functions f:[3]->[5]

The number of maps

f

from

1,2,3

into the set

1,2,3,4,5

such that

f(i)\le f(j)

whenever

i\le j

is

<span class='latex-bold'>(A)</span>~60

<span class='latex-bold'>(B)</span>~50

<span class='latex-bold'>(C)</span>~35

<span class='latex-bold'>(D)</span>~30

Problem 9

2

shape in complex plane

ABCD

is a quadrilateral on the complex plane whose four vertices satisfy

z^4+z^3+z^2+z+1=0

. Then

ABCD

is a

<span class='latex-bold'>(A)</span>~\text{Rectangle}

<span class='latex-bold'>(B)</span>~\text{Rhombus}

<span class='latex-bold'>(C)</span>~\text{Isosceles Trapezium}

<span class='latex-bold'>(D)</span>~\text{Square}

last digit of 2019^2019

Last digit of

2019^{2019}

is

Problem 8

1

sum of primes ≤1000

There are

168

primes below

1000

. Then sum of all primes below

1000

is,

<span class='latex-bold'>(A)</span>~11555

<span class='latex-bold'>(B)</span>~76127

<span class='latex-bold'>(C)</span>~57298

<span class='latex-bold'>(D)</span>~81722

Problem 7

1

number of six-digit squares with digits 123456

How many six-digit perfect squares can be formed using all the numbers

1,2,3,4,5,6

as digits?

<span class='latex-bold'>(A)</span>~5

<span class='latex-bold'>(B)</span>~19

<span class='latex-bold'>(C)</span>~7

<span class='latex-bold'>(D)</span>~\text{None of the above}

Problem 6

1

more liquid ratios

Vessel

A

has liquids

X

and

Y

in the ratio

X:Y=8:7

. Vessel

B

holds a mixture of

X

and

Y

in the ratio

X:Y=5:9

. What ratio should you mix the liquids in both vessels if you need the mixture to be

X:Y=1:1

?

<span class='latex-bold'>(A)</span>~4:3

<span class='latex-bold'>(B)</span>~30:7

<span class='latex-bold'>(C)</span>~17:25

<span class='latex-bold'>(D)</span>~7:30

Problem 5

2

arccos sum

If

\sum_{i=1}^n\cos^{-1}(\alpha_i)=0

, then find

\sum_{i=1}^n\alpha_i

.

<span class='latex-bold'>(A)</span>~\frac n2

<span class='latex-bold'>(B)</span>~n

<span class='latex-bold'>(C)</span>~n\pi

<span class='latex-bold'>(D)</span>~\frac{n\pi}2

placing numbers in chessboard

64

numbers (not necessarily distinct) are placed on the squares of a chessboard such that the sum of the numbers in every

2\times2

square is

7

. What is the sum of the four numbers in the corners of the board?

Problem 4

2

counting 5x5 grids

How many

5\times5

grids are possible such that each element is either

0

or

1

and each row sum and column sum is

4

?

<span class='latex-bold'>(A)</span>~64

<span class='latex-bold'>(B)</span>~32

<span class='latex-bold'>(C)</span>~120

<span class='latex-bold'>(D)</span>~96

tangents to circle from point

From a point

P

outside of a circle with centre

O

, tangent segments

\overline{PA}

and

\overline{PB}

are drawn. If

\frac1{\left|\overline{OA}\right|^2}+\frac1{\left|\overline{PA}\right|^2}=\frac1{16}

, then

\left|\overline{AB}\right|=

?

<span class='latex-bold'>(A)</span>~4

<span class='latex-bold'>(B)</span>~6

<span class='latex-bold'>(C)</span>~8

<span class='latex-bold'>(D)</span>~10

Problem 3

2

side inequality in triangle

In

\triangle ABC

,

\left|\overline{AB}\right|=\left|\overline{AC}\right|

,

D

is the foot of the perpendicular from

C

to

AB

and

E

the foot of the perpendicular from

B

to

AC

, then

<span class='latex-bold'>(A)</span>~\left|\overline{BC}\right|^3>\left|\overline{BD}\right|^3+\left|\overline{BE}\right|^3

<span class='latex-bold'>(B)</span>~\left|\overline{BC}\right|^3<\left|\overline{BD}\right|^3+\left|\overline{BE}\right|^3

<span class='latex-bold'>(C)</span>~\left|\overline{BC}\right|^3=\left|\overline{BD}\right|^3+\left|\overline{BE}\right|^3

<span class='latex-bold'>(D)</span>~\text{None of the above}

counting finite sequences

How many finite sequences

x_1,x_2,\ldots,x_m

are there such that

x_i=1

or

2

and

\sum_{i=1}^mx_i=10

?

<span class='latex-bold'>(A)</span>~89

<span class='latex-bold'>(B)</span>~73

<span class='latex-bold'>(C)</span>~107

<span class='latex-bold'>(D)</span>~119

Problem 2

2

cutting a square into an octagon

From a square with sides of length

2m

, corners are cut away so as to form a regular octagon. What is the area of the octagon in

m^2

?

<span class='latex-bold'>(A)</span>~2\sqrt3

<span class='latex-bold'>(B)</span>~\frac4{\sqrt3}

<span class='latex-bold'>(C)</span>~4\left(\sqrt2-1\right)

<span class='latex-bold'>(D)</span>~\text{None of the above}

computational of side lenght in quadrilateral

Let

ABCD

be a quadrilateral with sides

\left|\overline{AB}\right|=2

,

\left|\overline{BC}\right|=\left|\overline{CD}\right|=4

and

\left|\overline{DA}\right|=5

. The opposite angles,

\angle A

and

\angle C

are equal. The length of diagonal

BD

equals

<span class='latex-bold'>(A)</span>~2\sqrt6

<span class='latex-bold'>(B)</span>~3\sqrt3

<span class='latex-bold'>(C)</span>~3\sqrt6

<span class='latex-bold'>(D)</span>~2\sqrt3

Problem 1

2

ratio of liquids in can

A can contains a mixture of two liquids A and B in the ratio

7:5

. When

9

litres of the mixture are drawn and replaced by the same amount of liquid

B

, the ratio of

A

and

B

becomes

7:9

. How many litres of liquid A was contained in the can initially?

<span class='latex-bold'>(A)</span>~18

<span class='latex-bold'>(B)</span>~19

<span class='latex-bold'>(C)</span>~20

<span class='latex-bold'>(D)</span>~\text{None of the above}

polynomial interpolation

Let

p(x)

be a polynomial of degree

4

with leading coefficient

1

. Suppose

p(1)=1

,

p(2)=2

,

p(3)=3

and

p(4)=4

. Then

p(5)=

?

<span class='latex-bold'>(A)</span>~5

<span class='latex-bold'>(B)</span>~\frac{25}6

<span class='latex-bold'>(C)</span>~29

<span class='latex-bold'>(D)</span>~35

2019 LIMIT Category A

Subcontests

a!b!c!d!=24!

taxicab number

condition for z/(1-z^2)

given trigonometric equality, find angle

# of solns. to 3^x+4^x=8^x over R

number of nondecreasing functions f:[3]->[5]

shape in complex plane

last digit of 2019^2019

sum of primes ≤1000

number of six-digit squares with digits 123456

more liquid ratios

arccos sum

placing numbers in chessboard

counting 5x5 grids

tangents to circle from point

side inequality in triangle

counting finite sequences

cutting a square into an octagon

computational of side lenght in quadrilateral

ratio of liquids in can

polynomial interpolation

2019 LIMIT Category A

Subcontests

a!b!c!d!=24!

taxicab number

condition for z/(1-z^2)

given trigonometric equality, find angle

# of solns. to 3^x+4^x=8^x over R

number of nondecreasing functions f:[3]-&gt;[5]

shape in complex plane

last digit of 2019^2019

sum of primes &le;1000

number of six-digit squares with digits 123456

more liquid ratios

arccos sum

placing numbers in chessboard

counting 5x5 grids

tangents to circle from point

side inequality in triangle

counting finite sequences

cutting a square into an octagon

computational of side lenght in quadrilateral

ratio of liquids in can

polynomial interpolation

number of nondecreasing functions f:[3]->[5]

sum of primes ≤1000